An Operator-Asymptotic Approach to Periodic Homogenization for Equations of Linearized Elasticity

Abstract

We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength $|\chi |$ for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (“quasimomentum”) $\chi$ . We develop an asymptotic procedure in powers of $|\chi |$ , combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain $L^2\to L^2$ , $L^2\to H^1$ , and higher-order $L^2\to L^2$ norm-resolvent estimates in ${\mathbb{R}}^3$ . The $L^2\to H^1$ and higher-order $L^2\to L^2$ correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae.